The Independence of the Continuum Hypothesis

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The Independence of the Continuum Hypothesis

Overview

One of the questions that accompanied the rigorous foundation of set theory at the end of the nineteenth century was the relationship of the relative sizes of the set of real numbers and the set of rationals. The axioms that had been laid down shortly thereafter were expected to provide an answer to the question of whether there was any infinite number between the sizes of those two sets. After an earlier proof that there might be a negative answer to the question, the work of Paul J. Cohen in the 1960s demonstrated that the question was not settled by the standard axioms. As a result, the notion of truth for statements about infinite sets was regarded as perhaps in need of revision.

Background

The notion of the infinite was one of the legacies of Greek philosophy, as it appeared in the work of Aristotle (384-322 B.C.). Aristotle, however, dismissed the idea of the "actual infinite" in favor of the "potential infinite," a sequence that could be continued indefinitely. In the centuries that followed various writers took a look at the notion of the infinite, but the subject seemed to be hedged about with paradoxes—arguments that seemed to give opposing answers to a single question. Galileo (1564-1642) showed that the number of even numbers was the same as that of the number of whole numbers, which violated the principle that the whole was greater than the part. Mathematicians left the subject of the infinite to the philosophers, who speculated about it in vague and grandiose terms.

An immense change took place with the work of Georg Cantor (1845-1918). Although Cantor had a mystical streak, his writings were genuinely mathematical and consisted of arguments and proofs about the actual infinite. Cantor demonstrated that the number of rational numbers was the same as the number of whole numbers, which was itself something of a surprise, since there seem to be so many more fractions than whole numbers. What Cantor did was to define the notion of a one-to-one correspondence between two sets in terms of matching up each element in one set with one element in the other, and then argue that being able to find such a correspondence between two sets amounted to showing that they had the same number of members. Cantor went ahead to show that the set of real numbers was larger than the set of whole numbers by proving that there could be no oneto-one correspondence between them.

A fundamental question that Cantor could not answer was whether there were any infinite sets whose number of members lay in between the whole numbers and the real numbers. The claim that no such intermediate set existed was called the continuum hypothesis. The name comes from referring to the collection of real numbers in terms of a continuous sequence of points on a line. It was regarded as of sufficient importance that David Hilbert (1862-1943), in an address in 1900 that stated a mathematical agenda for the twentieth century, put the continuum hypothesis at the top of his list of problems.

In an effort to work on the continuum hypothesis and other problems in the theory of infinite sets, Ernst Zermelo (1871-1956) came up with a set of axioms to try to avoid any concealed paradoxes. Other mathematicians had already run aground on some of the intricacies of dealing with the infinite, and Zermelo's axiomatization (known in a slightly altered form as ZF) was designed both to avoid paradoxes and to make further progress possible. He had a particular interest in the status of what has become known as the axiom of choice, but his system of axioms for set theory proved to be useful in addressing other issues as well.

The most important advance in the first half of the twentieth century with regard to the continuum hypothesis was the work of Kurt Gödel (1906-1978). He had already established some of the most important results in mathematical logic and assured the field of its status as an independent discipline. Then in 1938 he proved that the continuum hypothesis was consistent with the ZF axioms for set theory. This result suggested that there was good reason to keep working on the problem with the hope that the continuum hypothesis could be proved from the axioms. On the other hand, it did not establish that the continuum hypothesis was a consequence of the axioms, which would have finally answered the question posed by Hilbert.

Paul J. Cohen (1934- ) was a mathematician who did not specialize in mathematical logic when he arrived at Stanford University as an assistant professor of mathematics in 1961. He did, however, have an impressive mathematical background, and he was looking for a problem of some importance on which to work. His attention was directed to the continuum hypothesis, and he undertook a thorough study of the literature that surrounded the earlier attempts to establish it on the basis of the axioms of set theory. Over the next few years he managed to create an entirely new technique in mathematical logic that enabled him to provide a kind of answer to Hilbert's question.

There are three possible relationships between a statement and a set of axioms: the statement can be provable from the axioms, its negation can be provable from the axioms, or the statement can be neither provable nor unprovable from the axioms. A good example is the parallel postulate included by Euclid (fl. 300 B.C.) in his list of axioms for geometry. For many years mathematicians tried to prove that statement on the basis of the other axioms provided by Euclid, but they were always unsuccessful. Not until the nineteenth century was it demonstrated that the parallel postulate could not be proved from the other axioms.

The way in which this was demonstrated was to come up with one model for the other axioms in which the parallel postulate was true and another in which the parallel postulate was false. A model is a specific collection of objects to which the axioms apply. If two different models for a set of axioms can give two different answers for the question of the truth of a statement, that statement is said to be independent of the axioms. Specific geometric models for the Euclidean axioms without the parallel postulate showed that the parallel postulate was indeed independent of the other axioms.

Gödel succeeded in showing that the continuum hypothesis was consistent with the axioms of set theory by constructing a model of set theory based on the axiom of constructibility. This is the assertion that every set is built up from other sets by certain well-defined processes. Within this model (called "the constructible universe") the continuum hypothesis could be proved. As a result, the continuum hypothesis had been shown to be consistent with the other axioms of set theory—in other words, no contradiction could arise from including the continuum hypothesis with the other axioms. There was some disagreement about whether the axiom of constructibility adequately captured mathematical intuition about the realm of infinite sets.

Cohen introduced the method of "forcing" to try to show the other side of independence for the continuum hypothesis. He needed a model for the other axioms of set theory in which the continuum hypothesis was false. From that it would follow that the continuum hypothesis was independent of the axioms of set theory. Forcing involves specifying which statements (Cohen started by working with statements about the positive whole numbers) are supposed to be true in the model being constructed. In particular, one introduces a kind of relationship that determines the truth of compound statements by the truth of the component statements of which it is made up. The only statements true in the model are those that are forced to be true by the forcing conditions. This kind of case-by-case analysis had also appeared in Gödel's proof that the continuum hypothesis was consistent with the axioms of set theory.

Impact

When the news of Cohen's result became known in the community of mathematical logicians, it was widely regarded as the most important development in set theory since it had first been axiomatized. He received the highest honor paid by the mathematical community when he received the Fields Medal at the International Mathematical Congress of 1966. The medal is given every four years to the outstanding mathematician under the age of 40. He was the first recipient of the medal for work in logic and helped to give the discipline an added boost in the judgment of the rest of the mathematical community.

The technique of forcing became a standard part of the repertoire of logicians working in the area of mathematics known as recursion theory. This field studies the complexity of mathematical subsets of the positive whole numbers and leads to a hierarchy of sets. Varieties of forcing have continued to be introduced in an effort to achieve more and more sophisticated models of the axioms for set theory. The term "Cohen reals" is used to refer to the numbers introduced by the stipulations of forcing conditions.

In a philosophical sense the issue of the truth of the continuum hypothesis has been a matter for much speculation in light of Cohen's result. If the continuum hypothesis is not implied by the standard axioms of set theory nor is its negation, then somehow the standard axioms of set theory leave open a rather fundamental issue about the relationship between the set of whole numbers and the set of the reals. Some philosophers of mathematics have urged the introduction of new axioms, especially those asserting the existence of extremely large infinite numbers, as a way of resolving the question. Others have suggested variants on the axiom of choice in which Zermelo was interested as more intuitive but capable of settling the truth of the continuum hypothesis. Still others have argued that there is no actual truth about statements of set theory, since the objects in question are so far removed from human intuition. The same kind of objection that had been raised to the axiom of constructibility was brought up with regard to the other proposed axioms as well. After the introduction by Cantor of the techniques that first led to the inclusion of infinite sets within the arsenal of mathematicians and not just philosophers, the demonstration by Cohen of the independence of the continuum hypothesis has taken the subject back into the domain for philosophers as well as mathematicians.

THOMAS DRUCKER

Further Reading

Bell, J.L. Boolean-valued Models and Independence Proofs in Set Theory. Oxford: Oxford University Press, 1977.

Cohen, Paul J. Set Theory and the Continuum Hypothesis. New York: W.S. Benjamin, 1966.

Gödel, Kurt. The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton: Princeton University Press, 1940.

Moore, Gregory H. Zermelo's Axiom of Choice: Its Origins,Development, and Influence. New York: Springer-Verlag, 1982.

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